Optimal control of an evolution hemivariational inequality involving history-dependent operators

Abstract In this paper we consider a class of feedback control systems described by an evolution hemivariational inequality involving history-dependent operators. Under the mild conditions, first, we prove a priori estimates of the solutions to the feedback control system. Then, an existence theorem for the feedback control system is obtained by using the well-known Bohnenblust-Karlin fixed point theorem. Moreover, we consider an optimal control problem driven by the feedback control system, and establish its solvability. Finally, a parabolic partial differential system with Clarke subgradient term is considered to illustrate the applicability of the theoretical results.

[1]  W. Han,et al.  Numerical analysis of an evolutionary variational–hemivariational inequality with application in contact mechanics☆ , 2017 .

[2]  Anna Ochal,et al.  Hemivariational inequalities for stationary Navier–Stokes equations , 2005 .

[3]  Mircea Sofonea,et al.  Numerical Analysis of Elliptic Hemivariational Inequalities , 2017, SIAM J. Numer. Anal..

[4]  Jiongmin Yong,et al.  Optimal Control Theory for Infinite Dimensional Systems , 1994 .

[5]  Nikolaos S. Papageorgiou.  On the Existence of Solutions for Nonlinear Parabolic Problems with Nonmonotone Discontinuities , 1997 .

[6]  Stanislaw Migórski,et al.  Optimal control for doubly nonlinear evolutionary inclusions , 2018, Appl. Math. Comput..

[7]  Mircea Sofonea,et al.  Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems , 2012 .

[8]  Shengda Zeng,et al.  A Class of time-fractional hemivariational inequalities with application to frictional contact problem , 2018, Commun. Nonlinear Sci. Numer. Simul..

[9]  Chong Li,et al.  Variational Inequalities for Set-Valued Vector Fields on Riemannian Manifolds: Convexity of the Solution Set and the Proximal Point Algorithm , 2012, SIAM J. Control. Optim..

[10]  Mircea Sofonea,et al.  A Class of Variational-Hemivariational Inequalities with Applications to Frictional Contact Problems , 2014, SIAM J. Math. Anal..

[11]  Dumitru Motreanu,et al.  Approximate Controllability for Nonlinear Evolution Hemivariational Inequalities in Hilbert Spaces , 2015, SIAM J. Control. Optim..

[12]  Maojun Bin,et al.  Relaxation in nonconvex optimal control for nonlinear evolution hemivariational inequalities , 2019 .

[13]  Valeri Obukhovskii,et al.  Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces , 2011 .

[14]  S. Migórski,et al.  On steady flow of non-Newtonian fluids with frictional boundary conditions in reflexive Orlicz spaces , 2018 .

[15]  Jen-Chih Yao,et al.  Existence of Solutions for Noncoercive Hemivariational Inequalities by an Equilibrium Approach Under Pseudomonotone Perturbation , 2014, J. Optim. Theory Appl..

[16]  Karl Kunisch,et al.  Optimal Control of Elliptic Variational–Hemivariational Inequalities , 2018, J. Optim. Theory Appl..

[17]  Shengda Zeng,et al.  Generalized Penalty and Regularization Method for Differential Variational-Hemivariational Inequalities , 2021, SIAM J. Optim..

[18]  S. Migórski,et al.  Stationary Oberbeck–Boussinesq model of generalized Newtonian fluid governed by multivalued partial differential equations , 2017 .

[19]  Zhenhai Liu,et al.  Relaxation in nonconvex optimal control problems described by fractional differential equations , 2014 .

[20]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[21]  Shengda Zeng,et al.  W1,p versus C1: The nonsmooth case involving critical growth , 2020 .

[22]  Shengda Zeng,et al.  Inverse problems for nonlinear quasi-variational inequalities with an application to implicit obstacle problems of p -Laplacian type , 2019, Inverse Problems.

[23]  Nikolas S. Papageorgiou,et al.  An introduction to nonlinear analysis , 2003 .

[24]  S. Migórski,et al.  A class of evolution variational inequalities with nonconvex constraints , 2018, Optimization.

[25]  S. Migórski,et al.  Stationary flow of non-Newtonian fluid with nonmonotone frictional boundary conditions , 2015 .

[26]  Stanislaw Migórski,et al.  Convergence analysis of numerical solutions for optimal control of variational-hemivariational inequalities , 2020, Appl. Math. Lett..

[27]  P. Nistri,et al.  Optimal feedback control for a semilinear evolution equation , 1994 .

[28]  Weimin Han,et al.  Numerical Analysis of a Hyperbolic Hemivariational Inequality Arising in Dynamic Contact , 2015, SIAM J. Numer. Anal..

[29]  Akhtar A. Khan,et al.  Inverse problems for nonlinear quasi-hemivariational inequalities with application to mixed boundary value problems , 2020, Inverse Problems.

[30]  Mircea Sofonea,et al.  Variational-Hemivariational Inequalities with Applications , 2017 .

[31]  S. Migórski,et al.  Well-posedness of history-dependent evolution inclusions with applications , 2019, Zeitschrift für angewandte Mathematik und Physik.

[32]  Mircea Sofonea,et al.  Advances in Variational and Hemivariational Inequalities , 2015 .

[33]  H. F. Bohnenblust,et al.  On a Theorem of Ville , 1949 .

[34]  Zhenhai Liu,et al.  Existence results for impulsive feedback control systems , 2019, Nonlinear Analysis: Hybrid Systems.

[35]  Akhtar A. Khan,et al.  Existence Theorems for Elliptic and Evolutionary Variational and Quasi-Variational Inequalities , 2015, J. Optim. Theory Appl..

[36]  Panagiotis D. Panagiotopoulos,et al.  Hemivariational Inequalities: Applications in Mechanics and Engineering , 1993 .

[37]  Zdzisław Denkowski,et al.  An Introduction to Nonlinear Analysis: Theory , 2013 .

[38]  Shengda Zeng,et al.  A class of fractional differential hemivariational inequalities with application to contact problem , 2018 .

[39]  Shengda Zeng,et al.  Nonlinear Quasi-hemivariational Inequalities: Existence and Optimal Control , 2021, SIAM J. Control. Optim..

[40]  Xiuwen Li,et al.  Sensitivity Analysis of Optimal Control Problems Described by Differential Hemivariational Inequalities , 2018, SIAM J. Control. Optim..

[41]  P. Panagiotopoulos.  Inequality problems in mechanics and applications , 1985 .

[42]  Mircea Sofonea,et al.  The Rothe Method for Variational-Hemivariational Inequalities with Applications to Contact Mechanics , 2016, SIAM J. Math. Anal..

[43]  P. D. Panagiotopoulos,et al.  Mathematical Theory of Hemivariational Inequalities and Applications , 1994 .

[44]  S. Migórski,et al.  Existence results for evolutionary inclusions and variational–hemivariational inequalities , 2015 .

[45]  Jen-Chih Yao,et al.  Lower Semicontinuity of the Solution Set to a Parametric Optimal Control Problem , 2012, SIAM J. Control. Optim..

[46]  J. Aubin,et al.  Differential inclusions set-valued maps and viability theory , 1984 .

[47]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[48]  W. A. Kirk,et al.  Asymptotically nonexpansive mappings , 1998 .

[49]  Shengda Zeng,et al.  Existence results for double phase implicit obstacle problems involving multivalued operators , 2020, Calculus of Variations and Partial Differential Equations.

[50]  S. Migórski,et al.  Evolutionary Oseen Model for Generalized Newtonian Fluid with Multivalued Nonmonotone Friction Law , 2018 .

[51]  M. Sofonea,et al.  Nonlinear Inclusions and Hemivariational Inequalities , 2013 .